I'm trying to prove the following inequalities are true, but they're a little too complex, is it even possible to prove that they are true? I suspect they are, but unless there is some simple method of proving it, I don't know if it's possible.
Where $0\leq \theta_{i,L}, \theta_{i,R} \leq \frac{\pi}{4}$ and $0 \leq \beta < \alpha <1 $. It's basically trying to show that if you decrease $\theta_{i,R}$ within the allowable range then the results of all 3 equations will decrease as well. In fact the inequalities may not be necessary, the main point is showing that the original equations (on the right side of each inequality) increase as $\theta_{i,R}$ increases, and decrease as $\theta_{i,R}$ decreases.